Interactions of multiple rhythms in a biophysical network of neurons

Neural oscillations, including rhythms in the beta1 band (12–20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequency, but very little is known about how such input affects the network’s behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs. We demonstrate that a cell has the ability to respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to that of the other. We show that this is a very general phenomenon, independent of the model used. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it.Published versio

New dynamics in cerebellar Purkinje cells: torus canards

We describe a transition from bursting to rapid spiking in a reduced mathematical model of a cerebellar Purkinje cell. We perform a slow-fast analysis of the system and find that -- after a saddle node bifurcation of limit cycles -- the full model dynamics follow temporarily a repelling branch of limit cycles. We propose that the system exhibits a dynamical phenomenon new to realistic, biophysical applications: torus canards.Comment: 4 pages; 4 figures (low resolution); updated following peer-review: language and definitions updated, Figures 1 and 4 updated, typos corrected, references added and remove

GABAB receptor-mediated, layer-specific synaptic plasticity reorganizes gamma-frequency neocortical response to stimulation

Repeated presentations of sensory stimuli generate transient gamma-frequency (30-80 Hz) responses in neocortex that show plasticity in a task-dependent manner. Complex relationships between individual neuronal outputs and the mean, local field potential (population activity) accompany these changes, but little is known about the underlying mechanisms responsible. Here we show that transient stimulation of input layer 4 sufficient to generate gamma oscillations induced two different, lamina-specific plastic processes that correlated with lamina-specific changes in responses to further, repeated stimulation: Unit rates and recruitment showed overall enhancement in supragranular layers and suppression in infragranular layers associated with excitatory or inhibitory synaptic potentiation onto principal cells, respectively. Both synaptic processes were critically dependent on activation of GABAB receptors and, together, appeared to temporally segregate the cortical representation. These data suggest that adaptation to repetitive sensory input dramatically alters the spatiotemporal properties of the neocortical response in a manner that may both refine and minimize cortical output simultaneously

Hetereogeneity in Neuronal Intrinsic Properties: A Possible Mechanism for Hub-Like Properties of the Rat Anterior Cingulate Cortex during Network Activity.

The anterior cingulate cortex (ACC) is vital for a range of brain functions requiring cognitive control and has highly divergent inputs and outputs, thus manifesting as a hub in connectomic analyses. Studies show diverse functional interactions within the ACC are associated with network oscillations in the β (20-30 Hz) and γ (30-80 Hz) frequency range. Oscillations permit dynamic routing of information within cortex, a function that depends on bandpass filter-like behavior to selectively respond to specific inputs. However, a putative hub region such as ACC needs to be able to combine inputs from multiple sources rather than select a single input at the expense of others. To address this potential functional dichotomy, we modeled local ACC network dynamics in the rat in vitro. Modal peak oscillation frequencies in the β- and γ-frequency band corresponded to GABAAergic synaptic kinetics as seen in other regions; however, the intrinsic properties of ACC principal neurons were highly diverse. Computational modeling predicted that this neuronal response diversity broadened the bandwidth for filtering rhythmic inputs and supported combination-rather than selection-of different frequencies within the canonical γ and β electroencephalograph bands. These findings suggest that oscillating neuronal populations can support either response selection (routing) or combination, depending on the interplay between the kinetics of synaptic inhibition and the degree of heterogeneity of principal cell intrinsic conductances.Wellcome Trus

Parietal low beta rhythm provides a dynamical substrate for a working memory buffer

Working memory (WM) is a component of the brain’s memory systems vital for interpretation of sequential sensory inputs and consequent decision making. Anatomically, WM is highly distributed over the prefrontal cortex (PFC) and the parietal cortex (PC). Here we present a biophysically detailed dynamical systems model for a WM buffer situated in the PC, making use of dynamical properties believed to be unique to this area. We show that the natural beta1 rhythm (12 to 20 Hz) of the PC provides a substrate for an episodic buffer that can synergistically combine executive commands (e.g., from PFC) and multimodal information into a flexible and updatable representation of recent sensory inputs. This representation is sensitive to distractors, it allows for a readout mechanism, and it can be readily terminated by executive input. The model provides a demonstration of how information can be usefully stored in the temporal patterns of activity in a neuronal network rather than just synaptic weights between the neurons in that network

Prefrontal oscillations modulate the propagation of neuronal activity required for working memory

[EN] Cognition involves using attended information, maintained in working memory (WM), to guide action. During a cognitive task, a correct response requires flexible, selective gating so that only the appropriate information flows from WM to downstream effectors that carry out the response. In this work, we used biophysically-detailed modeling to explore the hypothesis that network oscillations in prefrontal cortex (PFC), leveraging local inhibition, can independently gate responses to items in WM. The key role of local inhibition was to control the period between spike bursts in the outputs, and to produce an oscillatory response no matter whether the WM item was maintained in an asynchronous or oscillatory state. We found that the WM item that induced an oscillatory population response in the PFC output layer with the shortest period between spike bursts was most reliably propagated. The network resonant frequency (i.e., the input frequency that produces the largest response) of the output layer can be flexibly tuned by varying the excitability of deep layer principal cells. Our model suggests that experimentally-observed modulation of PFC beta-frequency (15-30 Hz) and gamma -frequency (30-80 Hz) oscillations could leverage network resonance and local inhibition to govern the flexible routing of signals in service to cognitive processes like gating outputs from working memory and the selection of rule-based actions. Importantly, we show for the first time that nonspecific changes in deep layer excitability can tune the output gate's resonant frequency, enabling the specific selection of signals encoded by populations in asynchronous or fast oscillatory states. More generally, this represents a dynamic mechanism by which adjusting network excitability can govern the propagation of asynchronous and oscillatory signals throughout neocortex.This work was supported by the U.S. Army Research Office under award number ARO W911NF-12-R-0012-02 to N. K., the U.S. Office of Naval Research under award number ONR MURI N00014-16-1-2832 to M. H. and E. M., the National Institute of Mental Health under award number NIMH R37MH087027 to E. M., and The MIT Picower Institute Faculty Innovation Fund to E. M. We would like to acknowledge Joachim Hass and Michelle McCarthy for early discussions of our modeling results, as well as Andre Bastos and Mikael Lundqvist for discussions relating our modeling work to their experiments.Sherfey, J.; Ardid-Ramírez, JS.; Miller, EK.; Hasselmo, ME.; Kopell, NJ. (2020). Prefrontal oscillations modulate the propagation of neuronal activity required for working memory. 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Flexible resonance in prefrontal networks with strong feedback inhibition

[EN] Oscillations are ubiquitous features of brain dynamics that undergo task-related changes in synchrony, power, and frequency. The impact of those changes on target networks is poorly understood. In this work, we used a biophysically detailed model of prefrontal cortex (PFC) to explore the effects of varying the spike rate, synchrony, and waveform of strong oscillatory inputs on the behavior of cortical networks driven by them. Interacting populations of excitatory and inhibitory neurons with strong feedback inhibition are inhibition-based network oscillators that exhibit resonance (i.e., larger responses to preferred input frequencies). We quantified network responses in terms of mean firing rates and the population frequency of network oscillation; and characterized their behavior in terms of the natural response to asynchronous input and the resonant response to oscillatory inputs. We show that strong feedback inhibition causes the PFC to generate internal (natural) oscillations in the beta/gamma frequency range (>15 Hz) and to maximize principal cell spiking in response to external oscillations at slightly higher frequencies. Importantly, we found that the fastest oscillation frequency that can be relayed by the network maximizes local inhibition and is equal to a frequency even higher than that which maximizes the firing rate of excitatory cells; we call this phenomenon population frequency resonance. This form of resonance is shown to determine the optimal driving frequency for suppressing responses to asynchronous activity. Lastly, we demonstrate that the natural and resonant frequencies can be tuned by changes in neuronal excitability, the duration of feedback inhibition, and dynamic properties of the input. Our results predict that PFC networks are tuned for generating and selectively responding to beta- and gamma-rhythmic signals due to the natural and resonant properties of inhibition-based oscillators. They also suggest strategies for optimizing transcranial stimulation and using oscillatory networks in neuromorphic engineering.This material is based upon research supported by the U. S. Army Research Office under award number ARO W911NF-12-R-0012-02 to N. K., the U. S. Office of Naval Research under award number ONR MURI N00014-16-1-2832 to M. H., and the National Science Foundation under award number NSF DMS-1042134 (Cognitive Rhythms Collaborative: A Discovery Network) to N. K. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Sherfey, JS.; Ardid-Ramírez, JS.; Hass, J.; Hasselmo, ME.; Kopell, NJ. (2018). Flexible resonance in prefrontal networks with strong feedback inhibition. 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A biophysical model of striatal microcircuits suggests gamma and beta oscillations interleaved at delta/theta frequencies mediate periodicity in motor control

Striatal oscillatory activity is associated with movement, reward, and decision-making, and observed in several interacting frequency bands. Local field potential recordings in rodent striatum show dopamine- and reward-dependent transitions between two states: a "spontaneous" state involving β (∼15-30 Hz) and low γ (∼40-60 Hz), and a state involving θ (∼4-8 Hz) and high γ (∼60-100 Hz) in response to dopaminergic agonism and reward. The mechanisms underlying these rhythmic dynamics, their interactions, and their functional consequences are not well understood. In this paper, we propose a biophysical model of striatal microcircuits that comprehensively describes the generation and interaction of these rhythms, as well as their modulation by dopamine. Building on previous modeling and experimental work suggesting that striatal projection neurons (SPNs) are capable of generating β oscillations, we show that networks of striatal fast-spiking interneurons (FSIs) are capable of generating δ/θ (ie, 2 to 6 Hz) and γ rhythms. Under simulated low dopaminergic tone our model FSI network produces low γ band oscillations, while under high dopaminergic tone the FSI network produces high γ band activity nested within a δ/θ oscillation. SPN networks produce β rhythms in both conditions, but under high dopaminergic tone, this β oscillation is interrupted by δ/θ-periodic bursts of γ-frequency FSI inhibition. Thus, in the high dopamine state, packets of FSI γ and SPN β alternate at a δ/θ timescale. In addition to a mechanistic explanation for previously observed rhythmic interactions and transitions, our model suggests a hypothesis as to how the relationship between dopamine and rhythmicity impacts motor function. We hypothesize that high dopamine-induced periodic FSI γ-rhythmic inhibition enables switching between β-rhythmic SPN cell assemblies representing the currently active motor program, and thus that dopamine facilitates movement in part by allowing for rapid, periodic shifts in motor program execution.R01 MH114877 - NIMH NIH HHSPublished versio

A biophysical model of striatal microcircuits suggests delta/theta - rhythmically interleaved gamma and beta oscillations mediate periodicity in motor control

First author draf

Mixed-Mode Oscillations in Three Time-Scale Systems: A Prototypical Example

Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) have been observed both experimentally and numerically in various prototypical systems in the natural sciences. In the present article, we propose a mathematical model problem which, though analytically simple, exhibits a wide variety of MMO patterns upon variation of a control parameter. One characteristic feature of our model is the presence of three distinct time-scales, provided a singular perturbation parameter is sufficiently small. Using geometric singular perturbation theory and geometric desingularization, we show that the emergence of MMOs in this context is caused by an underlying canard phenomenon. We derive asymptotic formulae for the return map induced by the corresponding flow, which allows us to obtain precise results on the bifurcation (Farey) sequences of the resulting MMO periodic orbits. We prove that the structure of these sequences is determined by the presence of secondary canards. Finally, we perform numerical simulations that show good quantitative agreement with the asymptotics in the relevant parameter regime